We first compute the partial derivatives of f(x, y):
f(x, y) = 5x² + 5y² - 10x + 21
∂f/∂x = 10x - 10
∂f/∂y = 10y
The critical points of f occur where both ∂f/∂x and ∂f/∂y are equal to zero. This only happens at the point (x, y) = (1, 0). (And this point does lie inside R.)
Next, compute the Hessian matrix H(x, y) for f :
[tex]H(x, y) = \begin{bmatrix}\frac{\partial^2f}{\partial x^2} & \frac{\partial^2f}{\partial x\partial y} \\ \frac{\partial^2 f}{\partial y\partial x} & \frac{\partial^2f}{\partial y^2}\end{bmatrix} = \begin{bmatrix}10&0\\0&10\end{bmatrix}[/tex]
Since det(H(x,y)) = 100 is positive for all x and y, this means the critical point (1, 0) is a local minimum, and we have f(1, 0) = 16.
Next, we check for extrema along the boundary of R, which is comprised of a semicircle with radius 2 and the line segment connecting (-2, 0) and (2, 0).
• Parameterize the semicircular portion by
x(t) = 2 cos(t)
y(t) = 2 sin(t)
with 0 ≤ t ≤ π. Then
f(x(t), y(t)) = 20 sin²(t) + 20 cos²(t) - 20 cos(t) + 21
which is simplifies to a function of one variable t,
g(t) = 41 - 20 cos(t)
Find the extrema of g over the interval [0, π] : we have critical points when
g'(t) = 20 sin(t) = 0
which happens at t = 0 and t = π. At these points, we get local extreme values of
•• t = 0 => x = 2 and y = 0 => f(2, 0) = 21
•• t = π => x = -2 and y = 0 => f(-2, 0) = 61
• Over the line-segment portion, we take y = 0, so f(x, y) again reduces to a function of one variable:
f(x, 0) = 5x² - 10x + 21
Completing the square, we have
5x² - 10x + 21 = 5 (x - 1)² + 16
which has a maximum value of 16 when x = 1 (and this happens to be the critical point (1, 0) we found earlier).
So, over the region R, f(x, y) has
• an absolute maximum of 61 at the point (-2, 0), and
• an absolute minimum of 16 at (1, 0)
Camila is collecting button pins. Camila started her collection with 20 button pins and collects 5 button pins per month. Let y represent the total number of button pins Camila has collected and x represent the number of months Camila collects button pins.
Graph this linear relationship.
Select two points to graph the line.
Step-by-step explanation:
Camilla started her collection with 20 buttons, meaning that this number is a constant, and that it will not change: (+20)
She then collects 5 buttons per month. Month is represented by the variable, x (given by the question), and there are 5 per month (so you multiply).
Set the equation:
Total pins = Amount of pins per month x amount of months + Amount of pins started.
y = 5x + 20
In this case, plug in the amount of months (x) to obtain your total (y):
When x = 0, y = 20
y = 5(0) + 20
y = 0 + 20
y = 20
When x = 1 , y = 25
y = 5(1) + 20
y = 5 + 20
y = 25
When x = 2, y = 30
y = 5(2) + 20
y = 10 + 20
y = 30
When x = 3, y = 35
y = 5(3) + 20
y = 15 + 20
y = 35
When x = 4, y = 40
y = 5(4) + 20
y = 20 + 20
y = 40
~
Note: This is a linear graph, so you can choose any of these given points (and more if you want to plot more). I would choose two that aren't too close to each other, unless you want a really small graph size.
What is the median of these figure skating ratings? 5.0 6.0 7.0 7.5 8.0 9.0 10.0
In a word processing document or on a separate piece of paper, use the guide to construct a two column proof proving that lines l and m are parallel. Angles 1 and 2 are supplementary by definition. Upload the entire proof below.
Given:
∠1 and ∠2 are supplementary angles
Prove:
l || m
STATEMENT REASON
1.∠1 and ∠2 are supplementary angles 1. Given
2. m∠1 + m∠2 = 180° 2.
3. ∠1 and ∠3 are supplementary angles 3. Exterior sides in opposite rays
4. 4.
5. m∠1 + m∠2 = m∠1 + m∠3 5.
6. 6.
7. l || m 7.
Line m and line l are parallel as m∠2 = m∠3
Equation
An equation is an expression that shows the relationship between two or more variables and numbers.
Two angles are said to be supplementary if they add up to 180 degrees.
m∠1 + m∠2 = 180° (Given)
m∠1 + m∠3 = 180° (sum of angles on a straight line)
Hence:
m∠1 + m∠2 = m∠1 + m∠3
m∠2 = m∠3
Line m and line l are parallel as m∠2 = m∠3
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9) You want to buy a backpack which costs $34.82. Now it is on sale of 30% off. What is the
discount amount? What is the final cost with the discount?
Answer:
10.446
Step-by-step explanation:
PLSSSS GUYS NOW FAST, This question has two parts. First, answer Part A. Then, answer Part B.
Part A
Adrian pays his niece $30 to babysit for 5 hours.
Which expression models the change in the amount of Adrian's cash each hour that his niece babysits?
Multiple choice question.
A)
30−5
B)
–30·5
C)
−30÷5
D)
–5÷30
Part B
What was the change in Adrian's cash each hour?
Multiple choice question.
A)
–$150
B)
–$6
C)
–$3
D)
$25
Answer:
Part A would be C because divided 30 by 5 to get the unit rate.
Part B would be B
Answer:
C). −30÷5
B). –$6
Step-by-step explanation:
A car rental agency charges $150 per week plus $0.20 per mile to rent a car. How many miles can
you
travel in one week for $290?
A) 208
B) 1450 miles
C) 700 miles
D) 675 miles
what is the solution of the equation ? 3=2w
Answer:
w = 1.5Step-by-step explanation:
Question:To find the solution that is 'w'Equation:3 = 2wSolution:3 = 2w
=> 2w = 3
[tex] = > w = \frac{3}{2} [/tex]
=> w = 1.5 (Ans)
If two fair dice are rolled, find the probability that the sum of the dice is 9, given that the sum is greater than 5.
The probability is
=====================================================
Explanation:
Refer to the dice chart below. It represents all possible ways to roll two dice and sum up the face values.
According to the table, there are...
A = 1 copy of the sum "2"B = 2 copies of the sum "3"C = 3 copies of the sum "4"D = 4 copies of the sum "5"There are A+B+C+D = 1+2+3+4 = 10 instances where we get a sum between 2 and 5. The table shows 6*6 = 36 sums total. This means there are 36-10 = 26 sums where they are between 6 and 12, aka "the sum being greater than 5".
Those 26 sums are what we focus on. Out of those 26 items, four of them are sums of "9".
The probability we want is therefore 4/26 = 2/13
The probability that the sum of the dice is 9, given that the sum is greater than 5 is 0.154
See attachment for the sample space of a roll of two dice.
From the attached sample space,
The number of sum greater than 5 is: 26The number of sum greater than 5 that is 9 is: 6Represent the events that a sum is greater than 5 with A, and a sum is greater than 5 is 9 with B.
So, we have:
[tex]n(A) = 26[/tex]
[tex]n(A\ n\ B) = 4[/tex]
The probability that the sum of the dice is 9, given that the sum is greater than 5 is then calculated using:
[tex]Pr = \frac{n(A\ n\ B)}{n(A)}[/tex]
This gives
[tex]Pr = \frac{4}{26}[/tex]
[tex]Pr = 0.154[/tex]
Hence, the probability is 0.154
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what is the lim of (8n)/((3n^2)+5)?
[tex]\lim \limits_{n \to \infty} ~\dfrac{8n}{3n^2 +5} \\\\\\=8\lim \limits_{n \to \infty} ~ \dfrac{\tfrac n{n^2}}{\tfrac{3n^2}{n^2} + \tfrac 5{n^2}}\\\\\\=8 \lim \limits_{n \to \infty} ~\dfrac{ \tfrac 1n}{ 3 + \tfrac 5{n^2}}}\\\\\\=8 \cdot \dfrac{ 0}{3 +0} = 8 \cdot 0 = 0[/tex]
We want to evaluate the limit of [tex]\frac{8n}{3n^{2}+5}[/tex] as n tends toward infinity (+∞)
The limit of this expression is the same as the limit of [tex]\frac{8n}{3n^{2}}[/tex]
But this last expression can be simplified, we can remove one n up and down which gives us : [tex]\frac{8}{3n}[/tex]
We know the limit of this one, it is simply Zero 0, by definition of the class lesson.
So to conclude the limit of [tex]\frac{8n}{3n^{2}+5}[/tex] as n tends toward infinity is 0.
Good Luck
Estimate the limit 1/x+2-1/4/x-1
Answer:
Step-by-step explanation:
Begin by finding a common denominator in the numerator of that rational expression. The common denominator is 3(x+2):
[tex]\frac{\frac{(3)1}{(3)x+2} -\frac{(x+2)1}{3(x+2)} }{x-1}[/tex] which simplifies to
[tex]\frac{\frac{3}{3x+6} -\frac{(x+2)}{3x+6} }{x-1}[/tex] which simplifies further to
[tex]\frac{\frac{3-x-2}{3x+6} }{x-1}[/tex] and
[tex]\frac{\frac{1-x}{3x+6} }{x-1}[/tex] Bring up the lower fraction and flip it and multiply:
[tex]\frac{1-x}{3x+6}*\frac{1}{x-1}[/tex] In order to make the 1-x into x-1, multiply the numerator of that fraction on the left by -1 to get:
[tex]-\frac{x-1}{(3x+6)(x-1)}[/tex] and now the x-1 terms cancel out, leaving us with:
[tex]-\frac{1}{3(1)+6}=-\frac{1}{9}=-.1111[/tex]
Find the equation of the line with slope −7 which goes through the point (−5,5).
Give your answer in slope-intercept form y=mx+b.
Answer:
[tex]\boxed {\boxed {\sf y= -7x -30}}[/tex]
Step-by-step explanation:
We are asked to find the equation of a line, given a point and a slope.
We can use the point-slope formula:
[tex]y-y_1= m(x-x_1)[/tex]
In this formula, (x₁, y₁) is the point and m is the slope. This line has a slope of -7 and passes through the point (-5,5). Therefore:
x₁ = -5 y₁= 5 m= -7Substitute the values into the formula.
[tex]y-5= -7 (x- -5)[/tex]
[tex]y-5= -7(x+5)[/tex]
We want to give the answer in slope-intercept form, or y=mx+b. We must isolate the variable y.
First, distribute the -7 on the right side of the equation. Multiply each term inside the parentheses by -7.
[tex]y-5= (-7*x) + (-7*5)\\y-5 = (-7x) + (-35) \\y-5= -7x -35[/tex]
5 is being subtracted from y. The inverse operation of subtraction is addition. Add 5 to both sides of the equation.
[tex]y-5+5 =-7x -35 +5 \\y= -7x -35 +5 \\y= -7x +30[/tex]
The equation of the line is y= -7x+30.
Which set of ordered pairs lies along the same vertical line?
Answer:
All ordered pairs with the same x value
Step-by-step explanation:
A vertical line can be represented by the equation x=k for some value k. So any pair with x value equal to k satisfies the equation and is thus in the set
3. the smallest number by which 168 must be multipled to get the perfect square number. ICA) 3 1(B)2 (c) 4 (o)5
Answer:
42
Step-by-step explanation:
168 factors into
2•2•2•3•7
two of the 2's make a perfect square
(2•2)•2•3•7
2²•2•3•7
as none of the product of the remaining factors makes a perfect square
2•3 = 6, 2•7 = 14, 3•7 = 21
we need to multiply the remaining factors together to find another perfect square
2²•(2•3•7)²
2•3•7 = 42
42(168) = 7056 = 84²
How do you do -9-(-2)-(-6)-4?
Coaching companies claim that their courses can raise the SAT scores of high school students. Of course, students who retake the SAT without paying for coaching generally raise their scores. A random sample of students who took the SAT twice found 427 who were coached and 2733 who were uncoached. Starting with their verbal scores on he first and second tries, we have these summary statistics:
Try 1 Try 2 Gain
n x¯¯¯ s x¯¯¯ s x¯¯¯ s
Coached 427 500 92 529 97 29 59
Uncoached 2733 506 101 527 101 21 52
Construct, using the conservative estimate for degrees of freedom, a 98% confidence interval for the difference, μc−μu between the mean gain of all students who are coached and the mean gain of all students who are uncoached.
Dеtеrmіnе tһе rаnցе օf tһе fսnсtіօn f(х) = 2х + 1 ԝһеn tһе ԁօmаіn іѕ {-3, 9, 15}.
Answer:
Range: {-5, 19, 31}
Step-by-step explanation:
Given the linear function, f(x) = 2x + 1, and the domain, {-3, 9, 15}.
In order to determine the range of the given linear function, simply substitute each domain value as inputs into the function.
x = -3f(x) = 2x + 1
f(-3) = 2(-3) + 1
f(-3) = -6 + 1
f(-3) = -5
x = 9f(9) = 2x + 1
f(9) = 2(9) + 1
f(9) = 18 + 1
f(9) = 19
x = 15f(15) = 2x + 1
f(15) = 2(15) + 1
f(15) = 30 + 1
f(15) = 31
Therfore, the range of the function is: {-5, 19, 31}.
need help with my math please <3
Answer:
wth that's hard I can't answer that
From 8:00 am to 8:20 am, Lisa picked 1/4 of a pound of strawberries. If she continues picking strawberries at this rate, how many pounds of strawberries will she pick in 4 hours?
Answer:
3 lbs
Step-by-step explanation:
20 mins is 1/4 lb (4oz)
1 hr is 3/4 lb (12 oz)
4 hrs= 3/4 lb or 12 oz × 4
4 hrs= 3 lbs or 48 oz
Write an equation of the line that is parallel to the y-axis and passes through the point (−3,1). Then determine whether the slope of the line is positive, negative, zero, or undefined.
An equation is
.
Question 2
The slope is positive.
The slope is positive.
The slope is negative.
The slope is negative.
The slope is zero.
The slope is zero.
The slope is undefined.
Answer:
The slope is undefined
Step-by-step explanation:
A vertical line has points that have the same x-coordinate.
The equation is x = -3
The slope of a vertical line is undefined.
Answer:
The slope is undefined
50 points! Help (Vectors)
[tex]2v - 3u\\\\=2<3,7> - 3<-5,4>\\\\=<6,14>-<-15,12>\\\\=<6+15, 14-12>\\\\=<21,2>[/tex]
Answer:
Step-by-step explanation:
The circle below represents one whole . …. What percent is represented by the shaded area ?
Answer:
50%
Step-by-step explanation:
shaded area = 2/4
shaded area as a percentage ;
2 / 4 x 100
= 1 / 2 x 100
= 1 / 1 x 50
= 50%
Select the correct answer from each drop-down menu.
A machine cuts a strip of carpet into two pleces. The length of the smaller plece is 5 meters greater than 4 the length of the larger piece. If
the length of the smaller piece is 12 meters, the length of the bigger plece is meters and the total length of the carpet is
meters.
Length of larger piece be x
Smaller piece=4x+5x=12m
[tex]\\ \sf\longmapsto 4x+5=4(12)+5=60+5=65m[/tex]
Total length=65+12=77m
The length of the bigger piece is 1.75 meters and the total length of the carpet is 13.75 meters.
Given that length of the smaller piece is 12 meters.
The length of the smaller piece is 5 meters greater than 4 times the length of the larger piece.
Let's represent the length of the larger piece as "x."
Length of the smaller piece = 4 times the length of the larger piece + 5
12 = 4x + 5
Subtracting 5 from both sides:
12 - 5 = 4x
7 = 4x
Dividing both sides by 4:
7/4 = x
1.75 = x
So, the length of the bigger piece is 1.75 meters.
To calculate the total length of the carpet, we add the length of the smaller piece and the length of the bigger piece:
Total length of the carpet = Length of the smaller piece + Length of the bigger piece
Total length of the carpet = 12 + 1.75
Total length of the carpet is approximately 13.75 meters.
Therefore, the length of the bigger piece is 1.75 meters and the total length of the carpet is 13.75 meters.
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2 short questions 20 POINTS
Step-by-step explanation:
13.
oh, come on !
you never had and solved the situation "one piece costs that much, how much will be 2 or 5 or 10 of them ?"
it is a simple multiplication.
and hence the answer B.
the first hint is already in the table with 1 pizza costing $5.25.
2 pizzas are then 2×$5.25, 3 pizzas 3×$5.25, ...
so, the cost c = number of pizzas × $5.25
or short
c = p×5.25 or 5.25p
14.
the domain is the interval or set of valid values for the input variable (x) of a function/relation.
and the range is the interval or set of valid values for the result variable (y) of a function/relation.
so, the domain is the set {-3, -1, 0, 1, 3}.
and the range is the set {-4, 0, 2, 4}.
In mathematics, a function is a binary relation between two intervals or sets that associates each element of the first interval/set (domain) to exactly one element of the second interval/set (range).
and that's what is done here.
it is a function.
each value of x is associated exactly with 1 result value of y.
a y value can be associated with more than one x value - no problem.
but no x value can have more than one associated y value - otherwise that would break the principle of a function.
Find the equation of the line shown below
Y=mx+c
Answer:
y = -1/4x + 2
Step-by-step explanation:
1. First you find m. Because the line is slanting downwards, you know it is negative. Then you can find the number by looking at the rise over run (rise/run). We can go up 1 (rise) and then to the side 4 (run). We go to the right side 4 because that is where the line intersects with up 1. So because the rise goes over the run, the number would be 1/4. But remember that it is negative so m = -1/4
2. Now we have to find c. For this you just have to go up or down the y-axis until you find the point where it intersects with the line. On this graph it intersects on positive 2. So c = + 2
Select two answers from below. x=_ and then which theorem justifies your answer.
(2x + 2)
(3x -63)
m
a
x=63
x=64
Ob
c
roob.
X=65
X=66
Alternate Interior Angles
Alternate Exterior Angles
Corresponding Angles
Same Side Interior Angles
f
Answer:
c. x=65
f. Alternate Exterior Angles
Step-by-step explanation:
2x+2=3x-63
2x=3x-65
-x=-65
x=65
They would be Alternate Exterior Angles because, they are 1, both on the outside of the parallel lines, and they are on different sides of the intersecting line, making them alternate. So, Alternate Exterior Angles
2. Michael Evans works a basic week of 40 hours. During a certain week he works 52 hours. If his weekly wage is $100.80 and overtime is paid at time and a half, what is his gross wage for the week? 3. In an engineering factory, employees work 38 hour basic Week. Any overtime worked from Monday to Friday is paid at time and half whilst overtime worked on Saturday and Sunday is paid at double time. If the basie wage is $2.48 per hour, find the gross ways for a man who works 12 hour overtime from Nlonday to Friday, 4 hours on Saturday and . 4 hours on Sunday
Answer: 3 :3 :3
$8.88
Step-by-step explanation:
(-5)⁶ ÷ (-5)⁸
Solution
A square piece of paper has a perimeter of 36 cm. What is the area of a square piece of paper with twice that perimeter? *
323 sq.cm
324 sq.cm
81 sq.cm
---------------------
( perimeter = l+ b )
(area = l × w )
(l= length)
(b= breadth )
(w= width)
Answer:
Area= 18 x 18=324 sq.cm
Step-by-step explanation:
p=36 x2=72 (as it is twice that of the original paper)
since it is a square, length of 1 side = 72/4= 18
so, the area = l x l= 18 x 18= 324 sq.cm
Answer:
Area= 18 x 18=324 sq.cm
Step-by-step explanation:
p=36 x2=72 (as it is twice that of the original paper)
Since it is a square, length of 1 side = 72/4= 18
Therefore, the area = l x l= 18 x 18= 324 sq.cm
Hope this helps :)
HURRY!! (No links please and please include an explanation)
If A is the set of all whole numbers, choose the set B that will make the following
statement false.
B ⊆ A
B = {0, 2, 4, 6)
B = {0}
B = {-1, 0, 1)
B = {1, 2, 3, 4)
Answer:
B= (-1,0,1)
Step-by-step explanation:
because -1 is not a whole number.
Answer:
B = {-1, 0, 1)
Step-by-step explanation:
The answer is B = {-1, 0, 1) because -1 is not a whole number.
Hope this helps :)
Help help help help help help help help help
Answer:
(-1,1)
Step-by-step explanation:
Im pretty sure the answer to A'' would be the inverse of A, as the triangle was mirrored on the opposite quadrant of the graph. (The inverse or opposite of (1,-1) would be (-1,1) ).
Hope this helped!